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An \(\tilde{O}(m^{2}n)\) Algorithm for Minimum Cycle Basis of Graphs

Abstract

We consider the problem of computing a minimum cycle basis of an undirected non-negative edge-weighted graph G with m edges and n vertices. In this problem, a {0,1} incidence vector is associated with each cycle and the vector space over \(\mathbb{F}_{2}\) generated by these vectors is the cycle space of G. A set of cycles is called a cycle basis of G if it forms a basis for its cycle space. A cycle basis where the sum of the weights of the cycles is minimum is called a minimum cycle basis of G. Minimum cycle basis are useful in a number of contexts, e.g. the analysis of electrical networks and structural engineering.

The previous best algorithm for computing a minimum cycle basis has running time O(m ω n), where ω is the best exponent of matrix multiplication. It is presently known that ω<2.376. We exhibit an O(m 2 n+mn 2log n) algorithm. When the edge weights are integers, we have an O(m 2 n) algorithm. For unweighted graphs which are reasonably dense, our algorithm runs in O(m ω) time. For any ε>0, we also design an 1+ε approximation algorithm. The running time of this algorithm is O((m ω/ε)log (W/ε)) for reasonably dense graphs, where W is the largest edge weight.

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Correspondence to Dimitrios Michail.

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A preliminary version of this paper appeared in Kavitha et al. (31st International Colloquium on Automata, Languages and Programming (ICALP), pp. 846–857, 2004).

T. Kavitha and K.E. Paluch were in Max-Planck-Institut für Informatik, Saarbrücken, Germany, while this work was done.

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Kavitha, T., Mehlhorn, K., Michail, D. et al. An \(\tilde{O}(m^{2}n)\) Algorithm for Minimum Cycle Basis of Graphs. Algorithmica 52, 333–349 (2008). https://doi.org/10.1007/s00453-007-9064-z

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Keywords

  • Cycle basis
  • Cycle space
  • Matrix multiplication
  • Polynomial algorithms